When a hammer strikes a nail, it is compressed along its length. Suppose that a steel nail 10.00 cm long and 3.00 mm in diameter is compressed an average of 65.0 µm by a 0.460 kg hammer during a 1.20 ms collision.
(a) What average force was exerted on the nail by the hammer?
(b) What is the impulse imparted to the hammer?
(c) If the hammer is moving at 7.00 m/s just before the collision, what is its final velocity?
To answer these questions, we can make use of several physics formulas and principles. Let's go step-by-step:
(a) To find the average force exerted on the nail, we can use Hooke's Law, which states that the force exerted by a solid object (in this case, the hammer) is proportional to the displacement it causes:
F = k * x
where F is the force, k is the stiffness constant (which depends on the material), and x is the displacement.
In this case, since the nail is being compressed, the constant k can be calculated using the Young's modulus formula:
k = (E * A) / L
where E is the Young's modulus of the material (for steel it is approximately 2 x 10^11 N/m^2), A is the cross-sectional area of the nail, and L is the original length of the nail.
Given:
L = 10.00 cm = 0.10 m (converting from cm to m)
Diameter = 3.00 mm = 3.00 x 10^(-3) m (converting from mm to m)
The cross-sectional area of the nail is:
A = π * (D/2)^2
= π * (3.00 x 10^(-3) / 2)^2
Now we can find k using the formula above.
Once we have k, we can use it along with the displacement x = 65.0 µm = 65.0 x 10^(-6) m to find the average force F.
(b) Impulse is equal to the change in momentum. Since momentum is given by the equation:
p = m * v
where p is momentum, m is mass, and v is velocity, we can calculate the impulse by subtracting the initial momentum from the final momentum.
Given:
Mass of the hammer, m = 0.460 kg
Initial velocity of the hammer, u = 7.00 m/s
Final velocity (which we need to find), v_f = ?
The impulse (J) is then given by:
J = m * (v_f - u)
(c) To find the final velocity of the hammer after the collision, we can use the impulse-momentum principle. It states that the impulse imparted to an object is equal to the change in its momentum.
Using the same formula as in part (b), we have:
J = m * (v_f - u)
Solving for v_f, we can find the final velocity of the hammer.
To summarize,
(a) Calculate the average force exerted on the nail using Hooke's Law.
(b) Calculate the impulse imparted to the hammer by subtracting the initial momentum from the final momentum.
(c) Calculate the final velocity of the hammer using the impulse-momentum principle.